A method for calculating the Painlevé transcendents

Open Problems for Painlevé Equations

“…Some recent numerical computations of Painlevé equations include: a pole field solver using Padé approximations [65,66,69,70,71,126,127]; numerical Riemann-Hilbert problems [119,120,122,121,133,142]; Fredholm determinants [26,27]; Padé approximations [110,112,144]; pole elimination [5,6,7,8,9,10,11]; a multidomain spectral method [96].…”

“…This is relatively simple to use, gives plots of solutions quickly with accuracy better than the human eye can detect, and generally works fine for initial value problems. Some recent numerical computations of Painlevé equations include: a pole field solver using Padé approximations [65,66,69,70,71,126,127]; numerical Riemann-Hilbert problems [119,120,122,121,133,142]; Fredholm determinants [26,27]; Padé approximations [110,112,144]; pole elimination [5,6,7,8,9,10,11]; a multidomain spectral method [96].…”

Numerical solution of Cauchy problems with multiple poles of integer order

“…Abramov and 106 DCM&ACS. 2022, 30 (2) [105][106][107][108][109][110][111][112][113][114] Yukhno proposed a special replacement for an unknown function that translates the solution into a non-singular one, see [7] and the bibliography there. However, these methods are applicable only for calculating the transcendental Painleves, for which there is a lot of a priori information.…”

Methods for the computation of the multivalued Painlevé transcendents on their Riemann surfaces

“…For a discussion of computational methods for the Painlevé equations that preceded the PFS, we refer to [9]. After the introduction of the PFS, another method for computing Painlevé transcendents was presented by Abramov and Yukhno [1]. Their method avoids singularities by making certain changes of variables in the neighborhoods of poles.…”

“…Their method avoids singularities by making certain changes of variables in the neighborhoods of poles. While [1] and the earlier methods, viz. the pole vaulting method [4] and the method in [30], should in theory be capable of computing Painlevé solutions over extended regions of the complex plane, no such results have been presented for either the multivalued or single-valued Painlevé transcendents, most likely due to a combination of cost and complexity.…”